- FindStat

Your data matches 83 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000256
(load all 2 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => 0 = 1 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => []
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [3]
 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => 0 = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [1]
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => []
 => 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => []
 => 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => []
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => []
 => 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => 0 = 1 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => []
 => 0 = 1 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => 0 = 1 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 0 = 1 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [1,1]
 => 0 = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => 0 = 1 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,1]
 => [1]
 => 0 = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => []
 => 0 = 1 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1]
 => 0 = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => []
 => 0 = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => []
 => 0 = 1 - 1

Description

The number of parts from which one can substract 2 and still get an integer partition.

Matching statistic: St000480
(load all 2 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => 0 = 1 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => []
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [3]
 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => 0 = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [1]
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => []
 => 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => []
 => 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => []
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => []
 => 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => 0 = 1 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => []
 => 0 = 1 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => 0 = 1 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 0 = 1 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [1,1]
 => 0 = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => 0 = 1 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,1]
 => [1]
 => 0 = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => []
 => 0 = 1 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1]
 => 0 = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => []
 => 0 = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => []
 => 0 = 1 - 1

Description

The number of lower covers of a partition in dominance order. According to [1], Corollary 2.4, the maximum number of elements one element (apparently for

$n\neq 2$ ) can cover is

$\frac{1}{2}(\sqrt{1+8n}-3)$ and an element which covers this number of elements is given by

$(c+i,c,c-1,\dots,3,2,1)$ , where

$1\leq i\leq c+2$ .

Matching statistic: St000183

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000183: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [2,2,2]
 => [2,2]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [4,1,1]
 => [1,1]
 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [4,1,1]
 => [1,1]
 => 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,1]
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => 1

Description

The side length of the Durfee square of an integer partition. Given a partition

$\lambda = (\lambda_1,\ldots,\lambda_n)$ , the Durfee square is the largest partition

$(s^s)$ whose diagram fits inside the diagram of

$\lambda$ . In symbols,

$s = \max\{ i \mid \lambda_i \geq i \}$ . This is also known as the Frobenius rank.

Matching statistic: St000533

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000533: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [2,2,2]
 => [2,2]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [4,1,1]
 => [1,1]
 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [4,1,1]
 => [1,1]
 => 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,1]
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => 1

Description

The minimum of the number of parts and the size of the first part of an integer partition. This is also an upper bound on the maximal number of non-attacking rooks that can be placed on the Ferrers board.

Matching statistic: St001432
(load all 2 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [2,2,2]
 => [2,2]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [4,1,1]
 => [1,1]
 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [4,1,1]
 => [1,1]
 => 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,1]
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => 1

Description

The order dimension of the partition. Given a partition

$\lambda$ , let

$I(\lambda)$ be the principal order ideal in the Young lattice generated by

$\lambda$ . The order dimension of a partition is defined as the order dimension of the poset

$I(\lambda)$ .

Matching statistic: St000257

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => []
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => []
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => []
 => 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [2]
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => []
 => []
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [2]
 => 0 = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => []
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [2]
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [3]
 => [1,1,1]
 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => []
 => 0 = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => []
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => []
 => []
 => 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => []
 => []
 => 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => []
 => []
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => []
 => []
 => 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,1]
 => [2]
 => 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [3]
 => 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => []
 => []
 => 0 = 1 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => []
 => 0 = 1 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,1]
 => [2]
 => 0 = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [2]
 => 0 = 1 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [3]
 => 0 = 1 - 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [1,1]
 => [2]
 => 0 = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => []
 => 0 = 1 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => [2]
 => 0 = 1 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => []
 => []
 => 0 = 1 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [1,1]
 => [2]
 => 0 = 1 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => []
 => []
 => 0 = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => []
 => []
 => 0 = 1 - 1

Description

The number of distinct parts of a partition that occur at least twice. See Section 3.3.1 of [2].

Matching statistic: St000481

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => []
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => []
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => []
 => 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [2]
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => []
 => []
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [2]
 => 0 = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => []
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [2]
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [3]
 => [1,1,1]
 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => []
 => 0 = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => []
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => []
 => []
 => 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => []
 => []
 => 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => []
 => []
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => []
 => []
 => 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,1]
 => [2]
 => 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [3]
 => 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => []
 => []
 => 0 = 1 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => []
 => 0 = 1 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,1]
 => [2]
 => 0 = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [2]
 => 0 = 1 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [3]
 => 0 = 1 - 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [1,1]
 => [2]
 => 0 = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => []
 => 0 = 1 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => [2]
 => 0 = 1 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => []
 => []
 => 0 = 1 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [1,1]
 => [2]
 => 0 = 1 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => []
 => []
 => 0 = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => []
 => []
 => 0 = 1 - 1

Description

The number of upper covers of a partition in dominance order.

Matching statistic: St000783
(load all 2 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000783: Integer partitions ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%

Values

([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [2,2,2]
 => [2,2]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [4,1,1]
 => [1,1]
 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [4,1,1]
 => [1,1]
 => 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,1]
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [13]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [13]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [13]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [13]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12,1]
 => [2,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,5),(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [13]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
 => [13]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [13]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,4),(0,6),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [13]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,5),(0,6),(1,2),(1,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [13]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,4),(0,6),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,5),(0,6),(1,4),(1,5),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,5),(0,6),(1,2),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,5),(0,6),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [13]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12,1]
 => [2,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [13]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,4),(0,6),(1,2),(1,3),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,4),(0,6),(1,2),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,4),(0,6),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,4),(0,6),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,3),(0,4),(0,6),(1,2),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,3),(0,4),(1,2),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,3),(0,4),(0,6),(1,2),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [13]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
([(0,3),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1

Description

The side length of the largest staircase partition fitting into a partition. For an integer partition

$(\lambda_1\geq \lambda_2\geq\dots)$ this is the largest integer

$k$ such that

$\lambda_i > k-i$ for

$i\in\{1,\dots,k\}$ . In other words, this is the length of a longest (strict) north-east chain of cells in the Ferrers diagram of the partition, using the English convention. Equivalently, this is the maximal number of non-attacking rooks that can be placed on the Ferrers diagram. This is also the maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram. A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic records the largest part occurring in any of these partitions.

Matching statistic: St000920
(load all 4 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%

Values

([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [2,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10,1]
 => [2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [13]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [13]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10,1]
 => [2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10,1]
 => [2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [9,1,1]
 => [3,1,1,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10,1]
 => [2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [10,1]
 => [2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10,1]
 => [2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,6),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [9,1,1]
 => [3,1,1,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [11,1]
 => [2,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,6),(1,5),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10,1]
 => [2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10,1]
 => [2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [13]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10,1]
 => [2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [10,1]
 => [2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [11,1]
 => [2,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,5),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [10,1]
 => [2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5)],7)
 => [10,1]
 => [2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [13]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
 => [11,1]
 => [2,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [11,1]
 => [2,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12,1]
 => [2,1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,5),(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [13]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
 => [13]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [13]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,6),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10,1]
 => [2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [8,3]
 => [2,2,2,1,1,1,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2
([(0,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10,1]
 => [2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,6),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10,1]
 => [2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,4),(0,6),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [10,1]
 => [2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [11,1]
 => [2,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1

Description

The logarithmic height of a Dyck path. This is the floor of the binary logarithm of the usual height increased by one:

$\lfloor\log_2(1+height(D))\rfloor$

Matching statistic: St001104

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001104: Dyck paths ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 100%

Values

([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => 1
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? = 1
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? = 1
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? = 1
([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [13]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? = 1
([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [12]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? = 1
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [12]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? = 1
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [13]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? = 1
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,4),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,5),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,5),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [7,3]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 2
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,6),(2,3),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [8,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [8,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? = 1
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,6),(2,3),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [8,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,2),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(0,6),(1,4),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1

Description

The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. Following Stembridge [1, cor.4.7], the highest weight words indexing the irreducibles in

$\mathfrak{gl_n}^{\otimes r}$ are ''staircase tableaux'' of length

$2r$ : sequences

$(\gamma^{(0)},\dots,\gamma^{(2r)})$ of vectors in

$\mathbb Z^n$ with decreasing entries, such that

$\gamma^{(2i+1)}$ is obtained from

$\gamma^{(2i)}$ by adding a unit vector and

$\gamma^{(2i)}$ is obtained from

$\gamma^{(2i-1)}$ by subtracting a unit vector. For

$n=2$ , the staircase tableaux whose final element is the zero vector are in natural correspondence with Dyck paths: adding the first or subtracting the second unit vector is translated to an up step, whereas adding the second or subtracting the first unit vector is translated to a down step. A Dyck path can be transformed into a ''bicoloured Motzkin path'' by replacing double up steps (double down, up-down, down-up steps) with up steps (down, coloured level, level steps). Note that the resulting path cannot have coloured level steps at height zero. In this context, say that a bicoloured Motzkin path has a

$\mathfrak{gl}_2$ -''descent'' between the following pairs of steps: * an up step followed by a level step * an up step followed by a down step, if the final height is not zero * a coloured level step followed by any non-coloured step. Then, conjecturally, the quasisymmetric expansion of the Frobenius character of the symmetric group

$\mathfrak S_r$ acting on

$\mathfrak{gl}_2^{\otimes r}$ , is

$\sum_M F_{Des(M)},$ where the sum is over all length

$r$ prefixes of bicoloured Motzkin paths,

$Des(M)$ is the set of indices of descents of the path

$M$ and

$F_D$ is Gessel's fundamental quasisymmetric function. The statistic recorded here is the number of

$\mathfrak{gl}_2$ -descents in the bicoloured Motzkin path corresponding to the Dyck path. Restricting to Motzkin paths without coloured steps one obtains the quasisymmetric expansion for the Frobenius character of

$\mathfrak S_r$ acting on

$\mathfrak{sl}_2^{\otimes r}$ . In this case, the conjecture was shown by Braunsteiner [2].

The following 73 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000660The number of rises of length at least 3 of a Dyck path. St000292The number of ascents of a binary word. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001712The number of natural descents of a standard Young tableau. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St001330The hat guessing number of a graph. St001871The number of triconnected components of a graph. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St000264The girth of a graph, which is not a tree. St001597The Frobenius rank of a skew partition. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St000455The second largest eigenvalue of a graph if it is integral. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001720The minimal length of a chain of small intervals in a lattice. St001618The cardinality of the Frattini sublattice of a lattice. St001845The number of join irreducibles minus the rank of a lattice. St001638The book thickness of a graph. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001890The maximum magnitude of the Möbius function of a poset. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000315The number of isolated vertices of a graph. St001651The Frankl number of a lattice. St001570The minimal number of edges to add to make a graph Hamiltonian. St000068The number of minimal elements in a poset. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001626The number of maximal proper sublattices of a lattice. St000699The toughness times the least common multiple of 1,. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000146The Andrews-Garvan crank of a partition. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000143The largest repeated part of a partition. St000185The weighted size of a partition. St000312The number of leaves in a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001214The aft of an integer partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition.